Fourier Analysis

Fourier Analysis

Translation by Olof Staffans of the lecture notes

Fourier analyysi

by

Gustaf Gripenberg

January 5, 2009

0

Contents

0 Integration theory 3

1 Finite Fourier Transform 10

1.1 Introduction . . . 10

1.2 L²-Theory (“Energy theory”) . . . 14

1.3 Convolutions (”Faltningar”) . . . 21

1.4 Applications . . . 31

1.4.1 Wirtinger’s Inequality . . . 31

1.4.2 Weierstrass Approximation Theorem . . . 32

1.4.3 Solution of Differential Equations . . . 33

1.4.4 Solution of Partial Differential Equations . . . 35

2 Fourier Integrals 36 2.1 L¹-Theory . . . 36

2.2 Rapidly Decaying Test Functions . . . 43

2.3 L²-Theory for Fourier Integrals . . . 45

2.4 An Inversion Theorem . . . 48

2.5 Applications . . . 52

2.5.1 The Poisson Summation Formula . . . 52

2.5.2 Is dL¹(R) = C⁰(R) ? . . . 53

2.5.3 The Euler-MacLauren Summation Formula . . . 54

2.5.4 Schwartz inequality . . . 55

2.5.5 Heisenberg’s Uncertainty Principle . . . 55

2.5.6 Weierstrass’ Non-Differentiable Function . . . 56

2.5.7 Differential Equations . . . 59

2.5.8 Heat equation . . . 63

2.5.9 Wave equation . . . 64 1

CONTENTS 2

3 Fourier Transforms of Distributions 67

3.1 What is a Measure? . . . 67

3.2 What is a Distribution? . . . 69

3.3 How to Interpret a Function as a Distribution? . . . 71

3.4 Calculus with Distributions . . . 73

3.5 The Fourier Transform of a Distribution . . . 76

3.6 The Fourier Transform of a Derivative . . . 77

3.7 Convolutions (”Faltningar”) . . . 79

3.8 Convergence in S^′ . . . 85

3.9 Distribution Solutions of ODE:s . . . 87

3.10 The Support and Spectrum of a Distribution . . . 89

3.11 Trigonometric Polynomials . . . 93

3.12 Singular differential equations . . . 95

4 Fourier Series 99 5 The Discrete Fourier Transform 102 5.1 Definitions . . . 102

5.2 FFT=the Fast Fourier Transform . . . 104

5.3 Computation of the Fourier Coefficients of a Periodic Function . . 107

5.4 Trigonometric Interpolation . . . 113

5.5 Generating Functions . . . 115

5.6 One-Sided Sequences . . . 116

5.7 The Polynomial Interpretation of a Finite Sequence . . . 118

5.8 Formal Power Series and Analytic Functions . . . 119

5.9 Inversion of (Formal) Power Series . . . 121

5.10 Multidimensional FFT . . . 122

6 The Laplace Transform 124 6.1 General Remarks . . . 124

6.2 The Standard Laplace Transform . . . 125

6.3 The Connection with the Fourier Transform . . . 126

6.4 The Laplace Transform of a Distribution . . . 129

6.5 Discrete Time: Z-transform . . . 130 6.6 Using Laguerra Functions and FFT to Compute Laplace Transforms131